Sparse approximation of triangular transports. Part I: the finite dimensional case
Abstract
For two probability measures $\rho$ and $\pi$ with analytic densities on the $d$dimensional cube $[1,1]^d$, we investigate the approximation of the unique triangular monotone KnotheRosenblatt transport $T:[1,1]^d\to [1,1]^d$, such that the pushforward $T_\sharp\rho$ equals $\pi$. It is shown that for $d\in\mathbb{N}$ there exist approximations $\tilde T$ of $T$, based on either sparse polynomial expansions or deep ReLU neural networks, such that the distance between $\tilde T_\sharp\rho$ and $\pi$ decreases exponentially. More precisely, we prove error bounds of the type $\exp(\beta N^{1/d})$ (or $\exp(\beta N^{1/(d+1)})$ for neural networks), where $N$ refers to the dimension of the ansatz space (or the size of the network) containing $\tilde T$; the notion of distance comprises the Hellinger distance, the total variation distance, the Wasserstein distance and the KullbackLeibler divergence. Our construction guarantees $\tilde T$ to be a monotone triangular bijective transport on the hypercube $[1,1]^d$. Analogous results hold for the inverse transport $S=T^{1}$. The proofs are constructive, and we give an explicit a priori description of the ansatz space, which can be used for numerical implementations.
 Publication:

arXiv eprints
 Pub Date:
 June 2020
 arXiv:
 arXiv:2006.06994
 Bibcode:
 2020arXiv200606994Z
 Keywords:

 Mathematics  Numerical Analysis;
 Mathematics  Statistics Theory;
 32D05;
 41A10;
 41A25;
 41A46;
 62D99;
 65D15
 EPrint:
 The original manuscript arXiv:2006.06994v1 has been split into two parts